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On the compactness of Julia sets of $$p$$-adic polynomials. (Sur la compacité des ensembles de Julia des polynômes $$p$$-adiques.) (French) Zbl 1047.37031
It is easy to see that the Julia set of a polynomial dynamical system over $$\mathbb C$$ is a compact subset of $$\mathbb P^1(\mathbb C)$$. If one considers $$\mathbb C_p$$ (the completion of an algebraic closure of the field of $$p$$-adic numbers) instead of $$\mathbb C$$, that is not necessarily so. The author shows that the compactness assumption imposes a strong restriction upon the dynamical system. In particular, if $$J(P_0)$$ is the Julia set of a polynomial $$P_0$$ of a degree $$\geq 2$$, and $$J(P_0)$$ is nonempty and compact, then all periodic points are repelling. In this case, the mapping $$P\mapsto J(P)$$ is continuous at $$P_0$$ with respect to the Hausdorff distance on the space of nonempty bounded closed subsets of $$\mathbb P^1(\mathbb C_p)$$.

##### MSC:
 37F50 Small divisors, rotation domains and linearization in holomorphic dynamics 11S85 Other nonanalytic theory 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
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